Allowing for space- and time-dependence of mass in Klein--Gordon equations resolves the problem of negative probability density and violation of Lorenz covariance of interaction in quantum mechanics. Moreover, it extends their applicability to the domain of quantum cosmology, where the variation in mass may be accompanied by high oscillations. In this paper, we propose a third-order exponential integrator, where the main idea lies in embedding the oscillations triggered by the possibly highly oscillatory component intrinsically into the numerical discretisation. While typically high oscillation requires appropriately small time steps, an application of Filon methods allows implementation with large time steps even in the presence of very high oscillation. This greatly improves the efficiency of the time-stepping algorithm. Proof of the convergence and its rate are nontrivial and require alternative representation of the equation under consideration. We derive careful bounds on the growth of global error in time discretisation and prove that, contrary to standard intuition, the error of time integration does not grow once the frequency of oscillations increases. Several of numerical simulations are presented to confirm the theoretical investigations and the robustness of the method in all oscillatory regimes.
翻译:允许Klein-Gordon方程式中质量的空间和时间依赖,可以解决负面概率密度和违反Lorenz在量子力学方面相互作用的偏差问题;此外,还将其适用性扩大到量子宇宙学领域,质量差异可能伴随着高振动。在本文件中,我们提议了一个第三阶指数集成器,其主要想法在于将可能高度悬浮的成分所引发的振荡部分嵌入数字离散中。虽然典型的高振荡需要适当的小时间步骤,但应用Filon方法可以使执行过程有较大的时间步骤,即使在非常高振荡的情况下也是如此。这大大提高了时间步算法的效率。这种趋同及其速度的证明是没有作用的,需要替代所考虑的方程。我们对时间离散中全球错误的增加进行了仔细的界限,并且证明,与标准直觉相反,时间融合的错误不会随着振荡频率的频率的增加而增加而增加。在几个数字模拟中提出了各种可靠的理论和模拟方法。